Low temperature behaviour of the random field Ising model

A relationship is established between the Ising model in a random positive and negative magnetic field on a lattice with p=¹₂. At T=0 each percolation cluster gives rise to a first-order transition at a different value of the field. There is a significant difference in behaviour between lattices for which p_c¹₂which have an infinite cluster, and p_c¹₂which do not. By considering the Bethe lattice and allowing the coordination number to become large, the results of the mean field approximation are reproduced. The above considerations do not apply to a Gaussian distribution of fields, and the absence of a first-order transition can be understood in this case. Since large clusters overturn for small fields there are clear indications of metastable behaviour.